
where \(\mathcal {B}(*{x})\)  is the transport efficiency or the inverse MacMullin number, while the tilde distinguishes the microscale (or bulk) value of the parameter from the macroscale (or effective) value of the parameter. In principle, \(\mathcal {B}(*{x})\)  could be a tensor, to account for the anisotropy of the porous matrix. Here, we treat the efficiency, \(\mathcal {B}(*{x})\) , as a placeholder and, in the literature, we find several options to model this factor; these range from an entirely systematic approach to computing this quantity from the exact microstructure [1]}, [2]} to ad hoc methods based on observed scalings of this factor with \(\varepsilon \)  (e.g. Bruggeman correlation [3]}, [4]} or Rayleigh method [5]}, [6]}). Interestingly, the systematic approach adopted in [1]}, [2]} can be shown to be equivalent to the approach adopted in [9]} in which this factor, which they write as \(\varepsilon /\tau \)  (where \(\tau \)  is termed the tortuosity factor), is computed from solution of Laplace's equation, with an appropriate boundary conditions, on a domain generated from tomography data of the electrode.
